As a new user - For home defense - you want a gun that is Reliable and Has a good warranty.
The S&W M&P15 (AR15) would be a good choice for you. Lifetime warranty. Very Reliable.
They are very reliable, in your cost range, and have a lifetime warranty(best in business).
I have 3 S&WM&P15's and can attest to their reliability and shootability. I also shoot Rifle Competitons every week
with mine and about 500rds/month in practice.
You "DO" need some training and practice on how to operate it efficiently - they are not as
easy to use/run as say a simple shotgun. Most AR15's are somewhat shorter than a shotgun and
thus easier to maneuver inside a house.
Many feel it is a better choice than a shotgun - a shotgun with buckshot will shoot thru as many walls as a 9mm pistol.
223 AR bullets start tumbling when shot thru sheetrock walls and can go thru fewer walls than Buckshot.
An AR15 has significantly less recoil than a shotgun - your wife/kids etc can shoot an AR15 easy - but maybe not a shotgun.
You may have to hand it to her or if she is home alone and needs it.
With "Any" gun fired in a house you need to watch your lanes of fire to be sure of what is beyond your target.

Ruger

If your set on a 223/5.56 for Home defense...
I’d buy a Ruger with a 10.5” barrel. Not sure they
Sell AR Pistols though? AR9 would b better...
Shotgun better than all with 18”-20” barrel!

For home protection, consider a Glock magazine compatible 9mm AR Pistol. There are several manufacturers out there now. With the 9mm AR, you still get your magazine capacity. You can get Glock 33 round magazines. You get the advantage of the use of red dot sites and, couple that with frangible type rounds, you won't have to worry about shooting through walls.
Freedom, DIamondback and Brigade all offer 9mm options. It is well worth looking at.

Best model I know for home defense is named Bertha, 300 pound biker chick.

A shotgun or pistol would probably be your best bet for home defence

I love my S&W sport II budsgunshop.com/product_info.php/products_id/4997/smith+%26+wesson+m%26p+15+sport+ii+223+rem%2f556+nato+16%22

John, there are loads in 5.56 and .223 that are suitable for home defense as well. I recommend Diamondback DB-15 as an excellent cost to performance ratio best buy. Unfortunately Buds is out of stock of all AR's in your price range. budsgunshop.com/product_info.php/products_id/100664/diamondback+firearms+db15+556+nato+16%22+black+w%2f12%22+m-lok+rail+30%2b1

For home protection in the house a shotgun with #4 shot works very well. A.223 slug will go through walls and could kill a family member, or a neighbor.